LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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zhetrs_3.f
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1*> \brief \b ZHETRS_3
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download ZHETRS_3 + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhetrs_3.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhetrs_3.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhetrs_3.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZHETRS_3( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB,
22* INFO )
23*
24* .. Scalar Arguments ..
25* CHARACTER UPLO
26* INTEGER INFO, LDA, LDB, N, NRHS
27* ..
28* .. Array Arguments ..
29* INTEGER IPIV( * )
30* COMPLEX*16 A( LDA, * ), B( LDB, * ), E( * )
31* ..
32*
33*
34*> \par Purpose:
35* =============
36*>
37*> \verbatim
38*> ZHETRS_3 solves a system of linear equations A * X = B with a complex
39*> Hermitian matrix A using the factorization computed
40*> by ZHETRF_RK or ZHETRF_BK:
41*>
42*> A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
43*>
44*> where U (or L) is unit upper (or lower) triangular matrix,
45*> U**H (or L**H) is the conjugate of U (or L), P is a permutation
46*> matrix, P**T is the transpose of P, and D is Hermitian and block
47*> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
48*>
49*> This algorithm is using Level 3 BLAS.
50*> \endverbatim
51*
52* Arguments:
53* ==========
54*
55*> \param[in] UPLO
56*> \verbatim
57*> UPLO is CHARACTER*1
58*> Specifies whether the details of the factorization are
59*> stored as an upper or lower triangular matrix:
60*> = 'U': Upper triangular, form is A = P*U*D*(U**H)*(P**T);
61*> = 'L': Lower triangular, form is A = P*L*D*(L**H)*(P**T).
62*> \endverbatim
63*>
64*> \param[in] N
65*> \verbatim
66*> N is INTEGER
67*> The order of the matrix A. N >= 0.
68*> \endverbatim
69*>
70*> \param[in] NRHS
71*> \verbatim
72*> NRHS is INTEGER
73*> The number of right hand sides, i.e., the number of columns
74*> of the matrix B. NRHS >= 0.
75*> \endverbatim
76*>
77*> \param[in] A
78*> \verbatim
79*> A is COMPLEX*16 array, dimension (LDA,N)
80*> Diagonal of the block diagonal matrix D and factors U or L
81*> as computed by ZHETRF_RK and ZHETRF_BK:
82*> a) ONLY diagonal elements of the Hermitian block diagonal
83*> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
84*> (superdiagonal (or subdiagonal) elements of D
85*> should be provided on entry in array E), and
86*> b) If UPLO = 'U': factor U in the superdiagonal part of A.
87*> If UPLO = 'L': factor L in the subdiagonal part of A.
88*> \endverbatim
89*>
90*> \param[in] LDA
91*> \verbatim
92*> LDA is INTEGER
93*> The leading dimension of the array A. LDA >= max(1,N).
94*> \endverbatim
95*>
96*> \param[in] E
97*> \verbatim
98*> E is COMPLEX*16 array, dimension (N)
99*> On entry, contains the superdiagonal (or subdiagonal)
100*> elements of the Hermitian block diagonal matrix D
101*> with 1-by-1 or 2-by-2 diagonal blocks, where
102*> If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
103*> If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
104*>
105*> NOTE: For 1-by-1 diagonal block D(k), where
106*> 1 <= k <= N, the element E(k) is not referenced in both
107*> UPLO = 'U' or UPLO = 'L' cases.
108*> \endverbatim
109*>
110*> \param[in] IPIV
111*> \verbatim
112*> IPIV is INTEGER array, dimension (N)
113*> Details of the interchanges and the block structure of D
114*> as determined by ZHETRF_RK or ZHETRF_BK.
115*> \endverbatim
116*>
117*> \param[in,out] B
118*> \verbatim
119*> B is COMPLEX*16 array, dimension (LDB,NRHS)
120*> On entry, the right hand side matrix B.
121*> On exit, the solution matrix X.
122*> \endverbatim
123*>
124*> \param[in] LDB
125*> \verbatim
126*> LDB is INTEGER
127*> The leading dimension of the array B. LDB >= max(1,N).
128*> \endverbatim
129*>
130*> \param[out] INFO
131*> \verbatim
132*> INFO is INTEGER
133*> = 0: successful exit
134*> < 0: if INFO = -i, the i-th argument had an illegal value
135*> \endverbatim
136*
137* Authors:
138* ========
139*
140*> \author Univ. of Tennessee
141*> \author Univ. of California Berkeley
142*> \author Univ. of Colorado Denver
143*> \author NAG Ltd.
144*
145*> \ingroup complex16HEcomputational
146*
147*> \par Contributors:
148* ==================
149*>
150*> \verbatim
151*>
152*> June 2017, Igor Kozachenko,
153*> Computer Science Division,
154*> University of California, Berkeley
155*>
156*> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
157*> School of Mathematics,
158*> University of Manchester
159*>
160*> \endverbatim
161*
162* =====================================================================
163 SUBROUTINE zhetrs_3( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB,
164 $ INFO )
165*
166* -- LAPACK computational routine --
167* -- LAPACK is a software package provided by Univ. of Tennessee, --
168* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
169*
170* .. Scalar Arguments ..
171 CHARACTER UPLO
172 INTEGER INFO, LDA, LDB, N, NRHS
173* ..
174* .. Array Arguments ..
175 INTEGER IPIV( * )
176 COMPLEX*16 A( LDA, * ), B( LDB, * ), E( * )
177* ..
178*
179* =====================================================================
180*
181* .. Parameters ..
182 COMPLEX*16 ONE
183 parameter( one = ( 1.0d+0,0.0d+0 ) )
184* ..
185* .. Local Scalars ..
186 LOGICAL UPPER
187 INTEGER I, J, K, KP
188 DOUBLE PRECISION S
189 COMPLEX*16 AK, AKM1, AKM1K, BK, BKM1, DENOM
190* ..
191* .. External Functions ..
192 LOGICAL LSAME
193 EXTERNAL lsame
194* ..
195* .. External Subroutines ..
196 EXTERNAL zdscal, zswap, ztrsm, xerbla
197* ..
198* .. Intrinsic Functions ..
199 INTRINSIC abs, dble, dconjg, max
200* ..
201* .. Executable Statements ..
202*
203 info = 0
204 upper = lsame( uplo, 'U' )
205 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
206 info = -1
207 ELSE IF( n.LT.0 ) THEN
208 info = -2
209 ELSE IF( nrhs.LT.0 ) THEN
210 info = -3
211 ELSE IF( lda.LT.max( 1, n ) ) THEN
212 info = -5
213 ELSE IF( ldb.LT.max( 1, n ) ) THEN
214 info = -9
215 END IF
216 IF( info.NE.0 ) THEN
217 CALL xerbla( 'ZHETRS_3', -info )
218 RETURN
219 END IF
220*
221* Quick return if possible
222*
223 IF( n.EQ.0 .OR. nrhs.EQ.0 )
224 $ RETURN
225*
226 IF( upper ) THEN
227*
228* Begin Upper
229*
230* Solve A*X = B, where A = U*D*U**H.
231*
232* P**T * B
233*
234* Interchange rows K and IPIV(K) of matrix B in the same order
235* that the formation order of IPIV(I) vector for Upper case.
236*
237* (We can do the simple loop over IPIV with decrement -1,
238* since the ABS value of IPIV(I) represents the row index
239* of the interchange with row i in both 1x1 and 2x2 pivot cases)
240*
241 DO k = n, 1, -1
242 kp = abs( ipiv( k ) )
243 IF( kp.NE.k ) THEN
244 CALL zswap( nrhs, b( k, 1 ), ldb, b( kp, 1 ), ldb )
245 END IF
246 END DO
247*
248* Compute (U \P**T * B) -> B [ (U \P**T * B) ]
249*
250 CALL ztrsm( 'L', 'U', 'N', 'U', n, nrhs, one, a, lda, b, ldb )
251*
252* Compute D \ B -> B [ D \ (U \P**T * B) ]
253*
254 i = n
255 DO WHILE ( i.GE.1 )
256 IF( ipiv( i ).GT.0 ) THEN
257 s = dble( one ) / dble( a( i, i ) )
258 CALL zdscal( nrhs, s, b( i, 1 ), ldb )
259 ELSE IF ( i.GT.1 ) THEN
260 akm1k = e( i )
261 akm1 = a( i-1, i-1 ) / akm1k
262 ak = a( i, i ) / dconjg( akm1k )
263 denom = akm1*ak - one
264 DO j = 1, nrhs
265 bkm1 = b( i-1, j ) / akm1k
266 bk = b( i, j ) / dconjg( akm1k )
267 b( i-1, j ) = ( ak*bkm1-bk ) / denom
268 b( i, j ) = ( akm1*bk-bkm1 ) / denom
269 END DO
270 i = i - 1
271 END IF
272 i = i - 1
273 END DO
274*
275* Compute (U**H \ B) -> B [ U**H \ (D \ (U \P**T * B) ) ]
276*
277 CALL ztrsm( 'L', 'U', 'C', 'U', n, nrhs, one, a, lda, b, ldb )
278*
279* P * B [ P * (U**H \ (D \ (U \P**T * B) )) ]
280*
281* Interchange rows K and IPIV(K) of matrix B in reverse order
282* from the formation order of IPIV(I) vector for Upper case.
283*
284* (We can do the simple loop over IPIV with increment 1,
285* since the ABS value of IPIV(I) represents the row index
286* of the interchange with row i in both 1x1 and 2x2 pivot cases)
287*
288 DO k = 1, n, 1
289 kp = abs( ipiv( k ) )
290 IF( kp.NE.k ) THEN
291 CALL zswap( nrhs, b( k, 1 ), ldb, b( kp, 1 ), ldb )
292 END IF
293 END DO
294*
295 ELSE
296*
297* Begin Lower
298*
299* Solve A*X = B, where A = L*D*L**H.
300*
301* P**T * B
302* Interchange rows K and IPIV(K) of matrix B in the same order
303* that the formation order of IPIV(I) vector for Lower case.
304*
305* (We can do the simple loop over IPIV with increment 1,
306* since the ABS value of IPIV(I) represents the row index
307* of the interchange with row i in both 1x1 and 2x2 pivot cases)
308*
309 DO k = 1, n, 1
310 kp = abs( ipiv( k ) )
311 IF( kp.NE.k ) THEN
312 CALL zswap( nrhs, b( k, 1 ), ldb, b( kp, 1 ), ldb )
313 END IF
314 END DO
315*
316* Compute (L \P**T * B) -> B [ (L \P**T * B) ]
317*
318 CALL ztrsm( 'L', 'L', 'N', 'U', n, nrhs, one, a, lda, b, ldb )
319*
320* Compute D \ B -> B [ D \ (L \P**T * B) ]
321*
322 i = 1
323 DO WHILE ( i.LE.n )
324 IF( ipiv( i ).GT.0 ) THEN
325 s = dble( one ) / dble( a( i, i ) )
326 CALL zdscal( nrhs, s, b( i, 1 ), ldb )
327 ELSE IF( i.LT.n ) THEN
328 akm1k = e( i )
329 akm1 = a( i, i ) / dconjg( akm1k )
330 ak = a( i+1, i+1 ) / akm1k
331 denom = akm1*ak - one
332 DO j = 1, nrhs
333 bkm1 = b( i, j ) / dconjg( akm1k )
334 bk = b( i+1, j ) / akm1k
335 b( i, j ) = ( ak*bkm1-bk ) / denom
336 b( i+1, j ) = ( akm1*bk-bkm1 ) / denom
337 END DO
338 i = i + 1
339 END IF
340 i = i + 1
341 END DO
342*
343* Compute (L**H \ B) -> B [ L**H \ (D \ (L \P**T * B) ) ]
344*
345 CALL ztrsm('L', 'L', 'C', 'U', n, nrhs, one, a, lda, b, ldb )
346*
347* P * B [ P * (L**H \ (D \ (L \P**T * B) )) ]
348*
349* Interchange rows K and IPIV(K) of matrix B in reverse order
350* from the formation order of IPIV(I) vector for Lower case.
351*
352* (We can do the simple loop over IPIV with decrement -1,
353* since the ABS value of IPIV(I) represents the row index
354* of the interchange with row i in both 1x1 and 2x2 pivot cases)
355*
356 DO k = n, 1, -1
357 kp = abs( ipiv( k ) )
358 IF( kp.NE.k ) THEN
359 CALL zswap( nrhs, b( k, 1 ), ldb, b( kp, 1 ), ldb )
360 END IF
361 END DO
362*
363* END Lower
364*
365 END IF
366*
367 RETURN
368*
369* End of ZHETRS_3
370*
371 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zswap(N, ZX, INCX, ZY, INCY)
ZSWAP
Definition: zswap.f:81
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:78
subroutine ztrsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
ZTRSM
Definition: ztrsm.f:180
subroutine zhetrs_3(UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO)
ZHETRS_3
Definition: zhetrs_3.f:165